/* Complex 16 bit Radix 4 FFT and Inverse FFT for Cortex-M3 version 1.0
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(c) 2008 Ivan Mellen 
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Free Personal, Non-Commercial Use License. 
The Software is licensed to you for your personal, NON-COMMERCIAL USE.
If you have questions about this license or would like a different license
please email : imellen(at)embeddedsignals(dot)com

- Radix 4 FFT - supported sizes: N=4,16,64,256,1024,4096
- N>4096 possible with custom coefficients 
- 16 bit complex arithmetic, 1Q15 coefficients
- input data remains unmodified
- decimation-in-time with auto scale after each stage - no overflow 
- GCC version (Code Sourcery G++ Lite 2007q3-53), requires C preprocessor
- hand optimized THUMB2 assembly for Cortex-M3 (e.g. STM32)
- code optimized with 64 bit flash prefetch and branch speculation in mind
- single function for multiple FFT sizes
- functions are both "C" and assembly callable

Modifications in version 2.0:
- Constant size FFT (faster execution, but N is constant)
- purely real input (almost 2x faster)
- radix 2 stage (for N= 8, 32, 128, 512, 2048, 8192...)
- RAM based coefficients with size optimized generator (for high flash latency)
- speed optimized windowing function(Bartlett,Blackman, Hamming, Hanning, Kaiser)


STM32 FFT benchmarks in CPU cycles based on real hardware measurements:

N - FFT size
L - Flash latency
F,R - coefficients in Flash or RAM

N L=0 F/R L=1 F L=1 r* L=2 F* L=2 r* ARM7TDMI ARM9E dsPIC
64 3575 3797 3636 4588 4007 - 2701 3739
256 19425 20685 19743 25144 21685 38461-43920 13740 19055
1024 98541 105113 100074 128070 109634 - 68534 N/A

Notes:ARM9E - DSP enhanced arm9 core, based on STR910 @96 MHz, RAM coefficients
dsPIC - dsPIC30x / dsPIC33x - results based on Microchip's DSP library 
ARM7TDMI - 3rd party, web based benchmark results
*STM32 results for latency L=2 or - source compiled with LATENCY2 defined

IFFT benchmarks (calculated and verified by measurement):
add 6+N/4 to FFT benchmark {22,70,262} for N={64,256,1024}

Code size:
FFT only: 480 bytes
FFT+IFFT: 682 bytes

Coefficient size (Flash or RAM):
N: 16 64 256 1024 4096
size: 48 240 1008 4080 16368 bytes

------------------------------------------------------------------------------
Usage example: add this file to your project. In C code:

//declare functions
void fftR4(short *y, short *x, int N);
void ifftR4(short *y, short *x, int N);

// prepare test input data
short x[512]; // input data 16 bit, 4 byte aligned x0r,x0i,x1r,x1i,....
short y[512]; // output data 16 bit,4 byte aligned y0r,y0i,y1r,y1i,....
short z[512]; // same format...

for (i=0;i<512;i++) x[i]=0;
for (i=0;i<512;i=i+8)
{ x[i+0]=16384; x[i+2]=16384;	x[i+4]=-16384; x[i+6]=-16384;}
// x = [ 16384,16384,-16384,-16384,16384,...] 1/4 Fsampling

//call functions
fftR4(y, x, 256); // y is in frequency domain y[128]=
ifftR4(z, y, 256); // z should be x/N + noise introduced by 16 bit truncating 


// expected results:
//y[128]is 8191; y[129] is -8190 y[384]is 8191; y[385]is 8191 rest 0 + noise
//z[2n] is 64 64 -64 -64 .. z[2n+1] is 0 all +- 1 (noise)
------------------------------------------------------------------------------
*/

// This file contains two functions :

// void fftR4(short *y, short *x, int N); // radix 4 FFT
// void ifftR4(short *y, short *x, int N); // radix 4 inverse FFT

	.syntax unified
	.thumb
	.global fftR4 
	.global ifftR4 

//#define LATENCY2 //comment this line if flash latency < 2 (Stelaris - leave commented)

#define y R0 // short *y -output complex array 
#define x R1 // short *x -input complex array 
#define N R2 // int N - number of samples 4,16,64,256,1024,4096

#define c R0 // short *c - coefficient array
#define Bl R2 // the number of blocks
#define R R3 // the number of samples in each block
#define x0r R4
#define x0i R5
#define x1r R6
#define x1i R7
#define x2r R8
#define x2i R9
#define x3r R10
#define x3i R11
#define y3r x3i
#define y3i x3r
#define tmp0 R12 // temporary0
#define tmp1 R14 // temporary1

// complex load, x=[a], a+=offset in register data from RAM (0 wait states)

.macro LOADC dr,di, a, offset //5 cycles
	ldrsh \di, [\a, #2]
	ldrsh \dr, [\a]	 //ldrsh $x._r, [$a], $offset
	adds \a, \offset
.endm

// cofficient complex load, c=[a], a=a+ 4 ;Wk from FLASH (1-4 wait states)
.macro LOADCF xr,xi, a //4 cycles
	ldr  \xr, [\a], #4	 //read 2 shorts as 32 bits
	asr  \xi, \xr,#16	 //im extract
	sxth \xr,\xr	 //re extract
.endm

// coefficient complex load, c=[a], a=a+ 4 ; Wk from RAM (0 wait states)
// .macro LOADCF xr,xi, a //4 (or 3?) cycles
// ldrsh xi, [$a, #2]
//	 ldrsh xr, [$a],#4 //ldrsh not pipelined in STM32, macro not used
// .endm

// complex load, xc=[a], a=a-register offset
.macro LOADCMi xr,xi, a, offset //5 cycles
	ldrsh \xi, [\a, #2]
	ldrsh \xr, [\a]
	subs  \a, \offset	 //ldrsh xr, [$a], offset
.endm

// complex store, [a]=x,c a=a+ immediate offset 
.macro STRC xr,xi, a, offset //2 cycles
	strh \xi, [\a, #2]
	strh \xr, [\a], \offset
.endm 

// complex store, [a]=x, a+=offset in register
.macro STRCR xr,xi, a, offset //3 cycles
	strh \xi, [\a, #2]
	strh \xr, [\a]
	adds \a, \offset
.endm 


// Multiply Complex Conjugate a=(xr+i*xi)*(cr-i*ci)
// x = xr + i*xi 
// c = cr + i*ci ;6 cycles
.macro MULCC ar,ai, xr,xi,cr,ci
	mul tmp0, \xi, \cr // tmp0=xi*cr
	mul tmp1, \xi, \ci // tmp1=xi*ci
	mls \ai, \xr, \ci, tmp0 // ai=tmp0-xr*ci = xi*cr - xr*ci
	mla \ar, \xr, \cr, tmp1 // ar=tmp1+xr*cr = xr*cr + xi*ci
.endm


// complex Fast Fourier Transform - Four point; scale with shift s
.macro BFFT4 s

	add x2r, x2r, x3r // (x2,x3) = (x2+x3, x2-x3)
	add x2i, x2i, x3i
	sub x3r, x2r, x3r, lsl#1
	sub x3i, x2i, x3i, lsl#1

	mov x0r, x0r, asr#2 // (x0,x1) = (x0+(x1>>s), x0-(x1>>s))/4
	mov x0i, x0i, asr#2
	add x0r, x0r, x1r, asr#(2+\s)
	add x0i, x0i, x1i, asr#(2+\s)
	sub x1r, x0r, x1r, asr#(1+\s)
	sub x1i, x0i, x1i, asr#(1+\s)

	add x0r, x0r, x2r, asr#(2+\s) // (x0,x2) = (x0+(x2>>s)/4, x0-(x2>>s)/4)
	add x0i, x0i, x2i, asr#(2+\s) 
	sub x2r, x0r, x2r, asr#(1+\s)
	sub x2i, x0i, x2i, asr#(1+\s)

	add x1r, x1r, x3i, asr#(2+\s) // (x1,y3)=(x1-i*(x3>>s)/4, x1+i*(x3>>s)/4)
	sub x1i, x1i, x3r, asr#(2+\s)
	sub y3r, x1r, x3i, asr#(1+\s)
	add y3i, x1i, x3r, asr#(1+\s)
.endm



//----------------------------------------------------------------------------
// inverse FFT 
// void ifftR4(short *y, short *x, int N) 
// custom first stage reorders input data: x(n)=x(N-n) n=0:N-1 x(N)=x(0)
// extra cost to fft: 6+N/4 cycles
	.thumb_func
	.align 3 //speed optimization in STM32
	nop.n //alignment optimization

ifftR4: stmfd sp!, {r4-r11, lr}
	mov tmp0, #0 // bit reversed counter
	movs tmp1,N //; first tmp1=N 
	.word 0xF3A2FA92 //RBIT R3,R2	//RBIT R,N
	lsl R,#3

	adds tmp1, x, tmp1, lsl#2 // tmp1=&x[tmp1==N] 
	ldrsh x0i, [x, #2] // replaces C_LDRmi x[N] by _LDRmi x[0] 
	ldrsh x0r, [x]
	subs tmp1,N //tmp1 still needs to be decremented for 2nd C_LDRmi
	b L2 // continue with second load

ifirstStage:
// first stage load and bit reverse
	adds tmp1, x, tmp1, lsl#2 // tmp1=&x[tmp1] 
	LOADCMi x0r,x0i, tmp1, N
L2:	LOADCMi x2r,x2i, tmp1, N
	LOADCMi x1r,x1i, tmp1, N
	LOADCMi x3r,x3i, tmp1, N
	BFFT4 0
	STRC x0r,x0i, y, #4
	STRC x1r,x1i, y, #4
	STRC x2r,x2i, y, #4
	STRC y3r,y3i, y, #4

	adds tmp0,R
	.word 0xFEACFA9C // rbit r14,r12//rbit tmp1,tmp0
	sub tmp1,N,tmp1 //tmp1=N-tmp1 
	bne ifirstStage // loop if count non zero 
	b firstStageFinished
// rest same as normal fft

//----------------------------------------------------------------------------
// void fftR4(short *y, short *x, int N)
	.thumb_func
fftR4:	stmfd sp!, {r4-r11, lr}
	mov tmp0, #0 // bit reversed counter
	mov tmp1, #0
	.word 0xF3A2FA92 //rbit R3,R2	//RBIT R,N
	lsl R,#3

firstStage:
// first stage load and bit reverse
	adds tmp1, x, tmp1, lsl#2 // tmp1=&x[tmp1] and clear carry
	LOADC x0r, x0i, tmp1, N
	LOADC x2r,x2i, tmp1, N
	LOADC x1r,x1i, tmp1, N
	LOADC x3r,x3i, tmp1, N
	BFFT4 0
	STRC x0r,x0i, y, #4
	STRC x1r,x1i, y, #4
	STRC x2r,x2i, y, #4
	STRC y3r,y3i, y, #4

	adds tmp0,R
	.word 0xFEACFA9C // rbit r14,r12//rbit tmp1,tmp0
	bne firstStage // loop if count non zero

firstStageFinished: // finished the first stage
	sub x, y, N, lsl #2 // x = working buffer
	mov R, #16
	lsrs Bl,N,4 
	it eq
	ldmeq sp!, {r4-r11, pc}	 // for N==4 return from function
	adr c, coef_table	//change if table in RAM 

nextStage:
// Bl = the number of blocks
// R = the number of samples in each block

#ifdef LATENCY2 // narrow/wide versions to optimize flash pre-fetch
	stm sp!, {x, Bl} 
	add tmp0, R, R, lsl#1
	add.n x, x, tmp0 
#else
	stmfd sp!, {x, Bl} 
	add tmp0, R, R, lsl#1
	add.w x, x, tmp0 
#endif
	sub Bl, Bl, #1<<16
nextBlock:
	add Bl, Bl, R, lsl#(16-2)

nextButterfly:
// Bl=((number butterflies left-1)<<16) + (number of blocks left)
	LOADCMi x0r,x0i, x, R
	LOADCF x3r,x3i, c
	MULCC x3r,x3i, x0r,x0i, x3r,x3i
	LOADCMi x0r,x0i, x, R
	LOADCF x2r,x2i, c
	MULCC x2r,x2i, x0r,x0i, x2r,x2i
	LOADCMi x0r,x0i, x, R
	LOADCF x1r,x1i, c
	MULCC x1r,x1i, x0r,x0i, x1r,x1i
	LOADC x0r,x0i, x, #0
	BFFT4 15 // coefficients are 1Q15
	STRCR x0r,x0i, x, R
	STRCR x1r,x1i, x, R
	STRCR x2r,x2i, x, R
	STRC y3r,y3i, x, #4
	subs Bl, Bl, #1<<16
	bge nextButterfly
	add tmp0, R, R, lsl#1
	add x, x, tmp0
	sub Bl, Bl, #1
	movs tmp1, Bl, lsl#16
	it ne
	subne c, c, tmp0
	bne nextBlock

	pop {r1-r2} //LDM sp!, {x, Bl}
	mov R, R, lsl#2 // block size *=4
	lsrs Bl,Bl,2 //# of blocks /=4
	bne nextStage
	ldmfd sp!, {r4-r11, pc}	 //return

	.align 2 //32 bit access acceleration

// Note: unused portion of coef_table can be commented to reduce size
coef_table:
// FFT twiddle table of triplets E(3t), E(t), E(2t)
// Where E(t)=cos(t)+i*sin(t) at 1Q15
// N=16 t=2*PI*k/N for k=0,1,2,..,N/4-1
	.hword 0x7fff,0x0000, 0x7fff,0x0000, 0x7fff,0x0000
	.hword 0x30fc,0x7642, 0x7642,0x30fc, 0x5a82,0x5a82
	.hword 0xa57e,0x5a82, 0x5a82,0x5a82, 0x0000,0x7fff
	.hword 0x89be,0xcf04, 0x30fc,0x7642, 0xa57e,0x5a82
// N=64 t=2*PI*k/N for k=0,1,2,..,N/4-1
	.hword 0x7fff,0x0000, 0x7fff,0x0000, 0x7fff,0x0000
	.hword 0x7a7d,0x2528, 0x7f62,0x0c8c, 0x7d8a,0x18f9
	.hword 0x6a6e,0x471d, 0x7d8a,0x18f9, 0x7642,0x30fc
	.hword 0x5134,0x62f2, 0x7a7d,0x2528, 0x6a6e,0x471d
	.hword 0x30fc,0x7642, 0x7642,0x30fc, 0x5a82,0x5a82
	.hword 0x0c8c,0x7f62, 0x70e3,0x3c57, 0x471d,0x6a6e
	.hword 0xe707,0x7d8a, 0x6a6e,0x471d, 0x30fc,0x7642
	.hword 0xc3a9,0x70e3, 0x62f2,0x5134, 0x18f9,0x7d8a
	.hword 0xa57e,0x5a82, 0x5a82,0x5a82, 0x0000,0x7fff
	.hword 0x8f1d,0x3c57, 0x5134,0x62f2, 0xe707,0x7d8a
	.hword 0x8276,0x18f9, 0x471d,0x6a6e, 0xcf04,0x7642
	.hword 0x809e,0xf374, 0x3c57,0x70e3, 0xb8e3,0x6a6e
	.hword 0x89be,0xcf04, 0x30fc,0x7642, 0xa57e,0x5a82
	.hword 0x9d0e,0xaecc, 0x2528,0x7a7d, 0x9592,0x471d
	.hword 0xb8e3,0x9592, 0x18f9,0x7d8a, 0x89be,0x30fc
	.hword 0xdad8,0x8583, 0x0c8c,0x7f62, 0x8276,0x18f9
// N=256 t=2*PI*k/N for k=0,1,2,..,N/4-1
	.hword 0x7fff,0x0000, 0x7fff,0x0000, 0x7fff,0x0000
	.hword 0x7fa7,0x096b, 0x7ff6,0x0324, 0x7fd9,0x0648
	.hword 0x7e9d,0x12c8, 0x7fd9,0x0648, 0x7f62,0x0c8c
	.hword 0x7ce4,0x1c0c, 0x7fa7,0x096b, 0x7e9d,0x12c8
	.hword 0x7a7d,0x2528, 0x7f62,0x0c8c, 0x7d8a,0x18f9
	.hword 0x776c,0x2e11, 0x7f0a,0x0fab, 0x7c2a,0x1f1a
	.hword 0x73b6,0x36ba, 0x7e9d,0x12c8, 0x7a7d,0x2528
	.hword 0x6f5f,0x3f17, 0x7e1e,0x15e2, 0x7885,0x2b1f
	.hword 0x6a6e,0x471d, 0x7d8a,0x18f9, 0x7642,0x30fc
	.hword 0x64e9,0x4ec0, 0x7ce4,0x1c0c, 0x73b6,0x36ba
	.hword 0x5ed7,0x55f6, 0x7c2a,0x1f1a, 0x70e3,0x3c57
	.hword 0x5843,0x5cb4, 0x7b5d,0x2224, 0x6dca,0x41ce
	.hword 0x5134,0x62f2, 0x7a7d,0x2528, 0x6a6e,0x471d
	.hword 0x49b4,0x68a7, 0x798a,0x2827, 0x66d0,0x4c40
	.hword 0x41ce,0x6dca, 0x7885,0x2b1f, 0x62f2,0x5134
	.hword 0x398d,0x7255, 0x776c,0x2e11, 0x5ed7,0x55f6
	.hword 0x30fc,0x7642, 0x7642,0x30fc, 0x5a82,0x5a82
	.hword 0x2827,0x798a, 0x7505,0x33df, 0x55f6,0x5ed7
	.hword 0x1f1a,0x7c2a, 0x73b6,0x36ba, 0x5134,0x62f2
	.hword 0x15e2,0x7e1e, 0x7255,0x398d, 0x4c40,0x66d0
	.hword 0x0c8c,0x7f62, 0x70e3,0x3c57, 0x471d,0x6a6e
	.hword 0x0324,0x7ff6, 0x6f5f,0x3f17, 0x41ce,0x6dca
	.hword 0xf9b8,0x7fd9, 0x6dca,0x41ce, 0x3c57,0x70e3
	.hword 0xf055,0x7f0a, 0x6c24,0x447b, 0x36ba,0x73b6
	.hword 0xe707,0x7d8a, 0x6a6e,0x471d, 0x30fc,0x7642
	.hword 0xdddc,0x7b5d, 0x68a7,0x49b4, 0x2b1f,0x7885
	.hword 0xd4e1,0x7885, 0x66d0,0x4c40, 0x2528,0x7a7d
	.hword 0xcc21,0x7505, 0x64e9,0x4ec0, 0x1f1a,0x7c2a
	.hword 0xc3a9,0x70e3, 0x62f2,0x5134, 0x18f9,0x7d8a
	.hword 0xbb85,0x6c24, 0x60ec,0x539b, 0x12c8,0x7e9d
	.hword 0xb3c0,0x66d0, 0x5ed7,0x55f6, 0x0c8c,0x7f62
	.hword 0xac65,0x60ec, 0x5cb4,0x5843, 0x0648,0x7fd9
	.hword 0xa57e,0x5a82, 0x5a82,0x5a82, 0x0000,0x7fff
	.hword 0x9f14,0x539b, 0x5843,0x5cb4, 0xf9b8,0x7fd9
	.hword 0x9930,0x4c40, 0x55f6,0x5ed7, 0xf374,0x7f62
	.hword 0x93dc,0x447b, 0x539b,0x60ec, 0xed38,0x7e9d
	.hword 0x8f1d,0x3c57, 0x5134,0x62f2, 0xe707,0x7d8a
	.hword 0x8afb,0x33df, 0x4ec0,0x64e9, 0xe0e6,0x7c2a
	.hword 0x877b,0x2b1f, 0x4c40,0x66d0, 0xdad8,0x7a7d
	.hword 0x84a3,0x2224, 0x49b4,0x68a7, 0xd4e1,0x7885
	.hword 0x8276,0x18f9, 0x471d,0x6a6e, 0xcf04,0x7642
	.hword 0x80f6,0x0fab, 0x447b,0x6c24, 0xc946,0x73b6
	.hword 0x8027,0x0648, 0x41ce,0x6dca, 0xc3a9,0x70e3
	.hword 0x800a,0xfcdc, 0x3f17,0x6f5f, 0xbe32,0x6dca
	.hword 0x809e,0xf374, 0x3c57,0x70e3, 0xb8e3,0x6a6e
	.hword 0x81e2,0xea1e, 0x398d,0x7255, 0xb3c0,0x66d0
	.hword 0x83d6,0xe0e6, 0x36ba,0x73b6, 0xaecc,0x62f2
	.hword 0x8676,0xd7d9, 0x33df,0x7505, 0xaa0a,0x5ed7
	.hword 0x89be,0xcf04, 0x30fc,0x7642, 0xa57e,0x5a82
	.hword 0x8dab,0xc673, 0x2e11,0x776c, 0xa129,0x55f6
	.hword 0x9236,0xbe32, 0x2b1f,0x7885, 0x9d0e,0x5134
	.hword 0x9759,0xb64c, 0x2827,0x798a, 0x9930,0x4c40
	.hword 0x9d0e,0xaecc, 0x2528,0x7a7d, 0x9592,0x471d
	.hword 0xa34c,0xa7bd, 0x2224,0x7b5d, 0x9236,0x41ce
	.hword 0xaa0a,0xa129, 0x1f1a,0x7c2a, 0x8f1d,0x3c57
	.hword 0xb140,0x9b17, 0x1c0c,0x7ce4, 0x8c4a,0x36ba
	.hword 0xb8e3,0x9592, 0x18f9,0x7d8a, 0x89be,0x30fc
	.hword 0xc0e9,0x90a1, 0x15e2,0x7e1e, 0x877b,0x2b1f
	.hword 0xc946,0x8c4a, 0x12c8,0x7e9d, 0x8583,0x2528
	.hword 0xd1ef,0x8894, 0x0fab,0x7f0a, 0x83d6,0x1f1a
	.hword 0xdad8,0x8583, 0x0c8c,0x7f62, 0x8276,0x18f9
	.hword 0xe3f4,0x831c, 0x096b,0x7fa7, 0x8163,0x12c8
	.hword 0xed38,0x8163, 0x0648,0x7fd9, 0x809e,0x0c8c
	.hword 0xf695,0x8059, 0x0324,0x7ff6, 0x8027,0x0648
// N=1024 t=2*PI*k/N for k=0,1,2,..,N/4-1
	.hword 0x7fff,0x0000, 0x7fff,0x0000, 0x7fff,0x0000
	.hword 0x7ffa,0x025b, 0x7fff,0x00c9, 0x7ffe,0x0192
	.hword 0x7fea,0x04b6, 0x7ffe,0x0192, 0x7ff6,0x0324
	.hword 0x7fce,0x0711, 0x7ffa,0x025b, 0x7fea,0x04b6
	.hword 0x7fa7,0x096b, 0x7ff6,0x0324, 0x7fd9,0x0648
	.hword 0x7f75,0x0bc4, 0x7ff1,0x03ed, 0x7fc2,0x07d9
	.hword 0x7f38,0x0e1c, 0x7fea,0x04b6, 0x7fa7,0x096b
	.hword 0x7ef0,0x1073, 0x7fe2,0x057f, 0x7f87,0x0afb
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